It is clear that, for a given input state, any problem of channel
discrimination collapses into a problem of state discrimination
[1]}, [2]}, [3]}, [4]}, [5]}, where we have to compute the
minimum error probability in identifying one of two possible output states. By
assuming \(M\)  identical copies of the input state and a memoryless quantum
channel, we have \(M\)  output states which are exact replicas of two
equiprobable states. In this case, the minimum error probability is
well-approximated by the quantum Chernoff bound (QCB) [6]}, [7]}, [8]}, [9]}, [10]}. Now the crucial step is to vary the input state,
trying to optimize the value of the output QCB. In the case of two lossy
bosonic channels, this kind of optimization must be constrained, meaning that
we have to fix some crucial parameters of the input states, in particular,
their energy. As we have already mentioned before, in our investigation we
optimize the QCB on the class of single- and two-mode squeezed thermal states
by fixing their total energy. Single-mode thermal states are sent through the
lossy channel, while two-mode thermal states probe the channel with the one of
the modes (probing mode) while bypassing the channel and assisting the output
measurement with the other mode (reference mode). In this scenario, we find
that the pure version of these states, i.e., single- and two-mode squeezed
vacuum states, are optimal for detecting losses, i.e., discriminating a lossy
from a lossless channel. Furthermore, we are able to show that, for any value
of the damping rate smaller than a certain critical value, there is a
threshold on the energy that makes the two-mode squeezed vacuum state more
convenient than the corresponding single-mode state. More interestingly, for
damping rates larger than the critical value, the two-mode squeezed vacuum
state performs always better than the single-mode squeezed vacuum state with
exactly the same energy.
